Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity

We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that low-rank QTT structure of the input is preserved in the output and propose efficient algorithms for these operations in the QTT format. For a d-dimensional 2n× . . .×2n-vector x given in a QTT representation with ranks bounded by p we show how a multilevel Toeplitz matrix generated by x can be obtained in the QTT format with ranks bounded by 2p in O ( dp log n ) operations. We also describe how the convolution x ? y of x and a d-dimensional n× . . .× n-vector y can be computed in the QTT format with ranks bounded by 2t in O ( dt log n ) operations, provided that the matrix xy′ is given in a QTT representation with ranks bounded by t. We exploit approximate matrix-vector multiplication in the QTT format to accelerate the convolution algorithm dramatically. We demonstrate high performance of the convolution algorithm with numerical examples including computation of the Newton potential of a strong cusp on fine grids with up to 2×2×2 points in 3D.